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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-328/3/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 328 / 3 / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Write

z = x + i y

and

\overline{z} = x - i y

. A convenient spectral parameter for a linear boundary value problem is any

λ \in C ∖ {0}

. Define the modified Helmholtz adjoint plane wave

v_{λ} (z, \overline{z}) = exp [\frac{ik}{2} (λ z - \frac{z}{λ})] = e^{A (λ) x + B (λ) y},

where

A (λ) = \frac{ik}{2} (λ - λ^{- 1}), B (λ) = - \frac{k}{2} (λ + λ^{- 1}) .

Indeed

A^{2} + B^{2} = k^{2}

, so

(Δ - k^{2}) v_{λ} = 0

. Equivalently,

4 \partial_{z} \partial_{\overline{z}} v_{λ} = k^{2} v_{λ}

. The exclusion of zero is only the coordinate singularity of this spectral parametrization; it still supplies a whole complex one-parameter family. The conjugate companion is

v_{λ} = exp [- \frac{ik}{2} (λ \overline{z} - \frac{z}{λ})] = e^{- A (λ) x + B (λ) y} = \overline{v_{\overline{λ}}} .

It also solves the adjoint equation.

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